Test 
Series 
Convergence or divergence 
Comments 
nth term test for divergence 

Diverges if 
Use this test first if divergence is suspected. However, test is
inconclusive if , so try another test. 
Geometric series 

(a) Converges if  r  < 1
(b) Diverges if

The first term is a and each term is a multiple, r of the
previous term. Also useful for comparison tests if the nth term of
a series is similar in magnitude to r ^{n1}. If
r< 1, . 
pseries 

(a) Converges if p > 1
(b) Diverges if 
Useful for comparison tests if the nth term of a series is similar
in magnitude to . 
Integral Test 

(a) Converges if converges
(b) Diverges if diverges 
The function ƒ obtained from must
be continuous, positive, decreasing. Use if you can integrate the
function. 
Individual Comparison
Test 
Given
compare to known
to converge or to diverge 
(a) If converges and
for every n then converges
(b) If diverges and
for every n then diverges 
The comparison series is
often a geometric series or a pseries.
To show convergence, you must find a series known to converge that is
greater than the given series.
To show divergence, you must find a series known to diverge that is
smaller than the given series.
Hints: ln n < n and sin n  and cos n
are always less than or equal to 1. 
Limit Comparison
Test 
, 
(a) If for some positive real number L then
both series converge or both diverge.
(b) If the limit is 0 and b_{n }is known to converge, then a_{n}
converges too.
(c) If the limit is ¥ and b_{n }is
known to diverge, a_{n} diverges too. 
To find b_{n} in the limit comparison test, consider only the
terms in a_{n} that have the greatest effect on the magnitude. Use
L’Hopital’s rule when needed. This often is helpful when ln n appears. 
Ratio
Test 

If , the series
(a) converges (absolutely) if L <1.
(b) diverges if L >1 (or ¥ ) 
Inconclusive if L = 1. Useful if a_{n} involves factorials or
nth powers. If all terms are positive then absolute value sign may be
disregarded.
(n+1)!/n! = n+1
2^{n+1}/2^{n} = 2
n +1 term for (2n)! is (2(n+1))! = (2n + 2)! 
Root Test 

If , the series
(a) converges (absolutely) if L <1.
(b) diverges if L >1 (or ¥ )

Inconclusive if L = 1. Useful if nth term involves nth powers. 
Alternating series
Test 

Converges if
(a)
(b) for every n >M
(c) 
Applies only to alternating series. Series diverges if any one of the
conditions is not met. Exponent on numeric factor may be n or n+1
depending on whether the terms are negative for n odd or even. 


If converges then converges
If a series converges absolutely, then it converges. A series that
converges but does not converge absolutely is called conditionally
convergent. 
Useful for series containing both positive and negative terms that do
not alternate. Use for series with trig functions. This is also useful
when you know the nonnegative series converges. Adding some negative
terms will only make the series converge more quickly. Note the converse
to this is not true. For example, the alternating harmonic series
converges by the alternating series test but the nonnegative harmonic
series diverges. 